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category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled directed edges are cal ...
, an abstract mathematical discipline, a nodal decomposition of a morphism \varphi:X\to Y is a representation of \varphi as a product \varphi=\sigma\circ\beta\circ\pi, where \pi is a strong epimorphism, \beta a bimorphism, and \sigma a strong monomorphism.A
monomorphism 220px In the context of abstract algebra or universal algebra, a monomorphism is an Injective function, injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of catego ...
\mu:C\to D is said to be strong, if for any
epimorphism 220px In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labe ...
\varepsilon:A\to B and for any morphisms \alpha:A\to C and \beta:B\to D such that \beta\circ\varepsilon=\mu\circ\alpha there exists a morphism \delta:B\to C, such that \delta\circ\varepsilon=\alpha and \mu\circ\delta=\beta


Uniqueness and notations

If exists, the nodal decomposition is unique up to an isomorphism in the following sense: for any two nodal decompositions \varphi=\sigma\circ\beta\circ\pi and \varphi=\sigma'\circ\beta'\circ\pi' there exist
isomorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s \eta and \theta such that : \pi'=\eta\circ\pi, : \beta=\theta\circ\beta'\circ\eta, : \sigma'=\sigma\circ\theta. This property justifies some special notations for the elements of the nodal decomposition: : \begin & \pi=\operatorname_\infty \varphi, && P=\operatorname_\infty \varphi,\\ & \beta=\operatorname_\infty \varphi, && \\ & \sigma=\operatorname_\infty \varphi, && Q=\operatorname_\infty \varphi, \end – here \operatorname_\infty \varphi and \operatorname_\infty \varphi are called the ''nodal coimage of \varphi'', \operatorname_\infty \varphi and \operatorname_\infty \varphi the ''nodal image of \varphi'', and \operatorname_\infty \varphi the ''nodal reduced part of \varphi''. In these notations the nodal decomposition takes the form :\varphi=\operatorname_\infty \varphi\circ\operatorname_\infty \varphi \circ \operatorname_\infty \varphi.


Connection with the basic decomposition in pre-abelian categories

In a pre-abelian category each morphism \varphi has a standard decomposition : \varphi=\operatorname \varphi\circ\operatorname \varphi\circ\operatorname \varphi, called the ''basic decomposition'' (here \operatorname \varphi=\ker(\operatorname \varphi), \operatorname \varphi=\operatorname(\ker\varphi), and \operatorname \varphi are respectively the image, the coimage and the reduced part of the morphism \varphi). If a morphism \varphi in a pre-abelian category has a nodal decomposition, then there exist morphisms \eta and \theta which (being not necessarily isomorphisms) connect the nodal decomposition with the basic decomposition by the following identities: : \operatorname_\infty \varphi=\eta\circ\operatorname \varphi, : \operatorname \varphi=\theta\circ\operatorname_\infty \varphi\circ\eta, : \operatorname_\infty \varphi=\operatorname \varphi\circ\theta.


Categories with nodal decomposition

A category is called a ''category with nodal decomposition'' if each morphism \varphi has a nodal decomposition in . This property plays an important role in constructing envelope (category theory), envelopes and refinement (category theory), refinements in . In an abelian category the basic decomposition : \varphi=\operatorname \varphi\circ\operatorname \varphi\circ\operatorname \varphi is always nodal. As a corollary, ''all abelian categories have nodal decomposition''. ''If a pre-abelian category is linearly complete, A category is said to be ''linearly complete'', if any functor from a linearly ordered set into has direct limit, direct and inverse limits. well-powered in strong monomorphismsA category is said to be ''well-powered in strong monomorphisms'', if for each object X the category \operatorname(X) of all strong monomorphisms into X is skeletally small (i.e. has a skeleton which is a set). and co-well-powered in strong epimorphisms,A category is said to be ''co-well-powered in strong epimorphisms'', if for each object X the category \operatorname(X) of all strong epimorphisms from X is skeletally small (i.e. has a skeleton which is a set). then has nodal decomposition.'' More generally, ''suppose a category is linearly complete, well-powered in strong monomorphisms, co-well-powered in strong epimorphisms, and in addition strong epimorphisms discern monomorphismsIt is said that ''strong epimorphisms discern monomorphisms'' in a category , if each morphism \mu, which is not a monomorphism, can be represented as a composition \mu=\mu'\circ\varepsilon, where \varepsilon is a strong epimorphism which is not an isomorphism. in , and, dually, strong monomorphisms discern epimorphismsIt is said that ''strong monomorphisms discern epimorphisms'' in a category , if each morphism \varepsilon, which is not an epimorphism, can be represented as a composition \varepsilon=\mu\circ\varepsilon', where \mu is a strong monomorphism which is not an isomorphism. in , then has nodal decomposition.'' The category Ste of stereotype spaces (being non-abelian) has nodal decomposition, as well as the (non-additive category, additive) category SteAlg of stereotype algebras .


Notes


References

* * *{{cite journal, last=Akbarov, first=S.S., title=Envelopes and refinements in categories, with applications to functional analysis, url=https://www.impan.pl/en/publishing-house/journals-and-series/dissertationes-mathematicae/all/513, journal=Dissertationes Mathematicae, year=2016, volume=513, pages=1–188, arxiv=1110.2013, doi=10.4064/dm702-12-2015, s2cid=118895911 Category theory